### Abstract

We considerably improve upon the recent result of [37] on the mixing time of Glauber dynamics for the 2D Ising model in a box of side L at low temperature and with random boundary conditions whose distribution P stochastically dominates the extremal plus phase. An important special case is when P is concentrated on the homogeneous all-plus configuration, where the mixing time TMIX is conjectured to be polynomial in L. In [37] it was shown that for a large enough inverse temperature and any " > 0 there exists c D c.; "/ such that limL!1P.TMIX exp.cL"// D 0. In particular, for the all-plus boundary conditions and large enough, TMIX exp.cL"/. Here we show that the same conclusions hold for all larger than the critical value c and with exp.cL"/ replaced by Lc logL (i.e. quasi-polynomial mixing). The key point is a modification of the inductive scheme of [37] together with refined equilibrium estimates that hold up to criticality, obtained via duality and random-line representation tools for the Ising model. In particular, we establish new precise bounds on the law of Peierls contours which complement the Brownian bridge picture established e.g. in [20, 22, 23].

Original language | English (US) |
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Pages (from-to) | 339-386 |

Number of pages | 48 |

Journal | Journal of the European Mathematical Society |

Volume | 15 |

Issue number | 2 |

DOIs | |

State | Published - Feb 4 2013 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

- Glauber dynamics
- Ising model
- Mixing time
- Phase coexistence

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## Cite this

*Journal of the European Mathematical Society*,

*15*(2), 339-386. https://doi.org/10.4171/JEMS/363